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Define the function $$f(q,z,y) = \sum_{n \ge 0,m,l} c(n,m,l) q^n z^m y^l$$ where $c(n,m,l)$ is defined by $$ c(n,m,l) = \begin{cases} (-1)^{s+l} & \text{if } 4n - m^2 + l^2 = 2s(s+1)\\ 0 & \text{otherwise} \end{cases} $$ for some integer $s$ and $c(n,m,l) = 0$ unless $4n - m^2 -l^2 \ge 0.$ $f(q,z,1)$ is known to be related to a Mock modular form. I conjecture that $$f(q,1,-1) = \sum_{n \ge 0} (-1)^n (2n + 1) q^{n(n+1)/2}.$$ Is there an elementary proof of the above conjecture? Is the function $f(q,z,y)$ a known mathematical object, perhaps related to a Siegel modular form?

Update: $f(q,z,y)$ is a product of Jacobi theta functions and $\mu(q;z,y),$ where $\mu(q;z,y)$ is a Lerch sum studied by Zweger in his thesis. Zweger's thesis also relates mock modular forms to indefinite quadratic forms of signature $(1,n),$ so perhaps it isn't too unsurprising that $f(q,z,y)$ takes a "nice" form.

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Why isn't there any $z$ dependence on the rhs of your equation for $f(q,z,-1)$? Also, can you provide details on the relation between f(q,z,1) and a specific mock modular form? – Jeff Harvey Jan 30 '13 at 17:26
@Jeff Harvey - Thanks! Typo fixed and (some) details added. – Richard Eager Jan 31 '13 at 9:50
I don't know the answer to your question, but one place that Appell-Lerch sums show up is in the Polar part of meromorphic Jacobi forms. The recent paper arXiv:1208.4074 by Dabholkar, Murthy and Zagier has a detailed treatment of the relation between meromorphic Jacobi forms and mock modular forms along with many examples. – Jeff Harvey Jan 31 '13 at 20:19

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